Select The Correct Answer.Consider These Functions:${ \begin{array}{l} f(x)=x+1 \ g(x)=\frac{2}{x} \end{array} }$Which Polynomial Is Equivalent To { (f \cdot G)(x)$}$?A. { \frac{2}{x+1}$}$B.

Mar 13, 2025 by ADMIN 192 views

**Function Composition and Polynomial Equivalence**

Understanding Function Composition

In mathematics, function composition is a process of combining two or more functions to create a new function. Given two functions, f(x) and g(x), the composition of f and g, denoted as (f ∘ g)(x) or (f ⋅ g)(x), is defined as:

(f ∘ g)(x) = f(g(x))

In this article, we will explore the composition of two given functions, f(x) = x + 1 and g(x) = 2/x, and determine which polynomial is equivalent to (f ⋅ g)(x).

Function Composition of f(x) and g(x)

To find the composition of f and g, we need to substitute g(x) into f(x) in place of x. This gives us:

(f ∘ g)(x) = f(g(x)) = f(2/x) = (2/x) + 1

Simplifying the Composition

Now, let's simplify the composition by finding a common denominator for the terms:

(f ∘ g)(x) = (2/x) + 1 = (2 + x)/x

Polynomial Equivalence

We are asked to determine which polynomial is equivalent to (f ⋅ g)(x). Let's examine the options:

A. 2/(x + 1) B. (2 + x)/x C. x/(2 + x) D. (x + 1)/2

Comparing Options

Comparing the simplified composition (f ∘ g)(x) = (2 + x)/x with the options, we can see that option B is the correct answer.

Conclusion

In conclusion, the polynomial equivalent to (f ⋅ g)(x) is (2 + x)/x. This is obtained by simplifying the composition of f(x) = x + 1 and g(x) = 2/x.

Final Answer

The final answer is B.

Understanding Function Composition

(f ∘ g)(x) = f(g(x))

In this article, we will explore the composition of two given functions, f(x) = x + 1 and g(x) = 2/x, and determine which polynomial is equivalent to (f ⋅ g)(x). We will also address some common questions and concerns related to function composition and polynomial equivalence.

Q&A: Function Composition and Polynomial Equivalence

Q: What is function composition?

A: Function composition is a process of combining two or more functions to create a new function. Given two functions, f(x) and g(x), the composition of f and g, denoted as (f ∘ g)(x) or (f ⋅ g)(x), is defined as:

(f ∘ g)(x) = f(g(x))

Q: How do I find the composition of two functions?

A: To find the composition of two functions, f(x) and g(x), you need to substitute g(x) into f(x) in place of x. This gives you:

(f ∘ g)(x) = f(g(x)) = f(2/x) = (2/x) + 1

Q: What is the difference between (f ∘ g)(x) and (f ⋅ g)(x)?

A: (f ∘ g)(x) and (f ⋅ g)(x) are both notations for function composition. They are often used interchangeably, but (f ∘ g)(x) is more commonly used in mathematics.

Q: How do I simplify the composition of two functions?

A: To simplify the composition of two functions, you need to find a common denominator for the terms. In the case of (f ∘ g)(x) = (2 + x)/x, the common denominator is x.

Q: What is the final answer to the problem?

A: The final answer to the problem is B. (2 + x)/x.

Q: Can you provide more examples of function composition?

A: Yes, here are a few more examples of function composition:

(f ∘ g)(x) = f(g(x)) = f(2x) = 2x + 1
(f ∘ g)(x) = f(g(x)) = f(x^2) = x^2 + 1
(f ∘ g)(x) = f(g(x)) = f(1/x) = 1/x + 1

Q: How do I determine which polynomial is equivalent to (f ⋅ g)(x)?

A: To determine which polynomial is equivalent to (f ⋅ g)(x), you need to compare the simplified composition with the options. In this case, the correct answer is B. (2 + x)/x.

Conclusion

In conclusion, function composition is a process of combining two or more functions to create a new function. Given two functions, f(x) and g(x), the composition of f and g, denoted as (f ∘ g)(x) or (f ⋅ g)(x), is defined as:

(f ∘ g)(x) = f(g(x))

We have explored the composition of two given functions, f(x) = x + 1 and g(x) = 2/x, and determined which polynomial is equivalent to (f ⋅ g)(x). We have also addressed some common questions and concerns related to function composition and polynomial equivalence.

Final Answer

The final answer is B.